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T TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.:

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Presentation on theme: "T TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.:"— Presentation transcript:

1 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 0 The diagram summarizes the procedure for performing a 5% significance test on the slope coefficient of a regression under the assumption that we know its standard deviation. 1 5% significance test: reject H 0 : 2 = 2 if z > 1.96 or z < –1.96

2 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.: s.d. of b 2 not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 0 This is a very unrealistic assumption. We usually have to estimate it with the standard error, and we use this in the test statistic instead of the standard deviation. 2 5% significance test: reject H 0 : 2 = 2 if z > 1.96 or z < –1.96

3 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.: s.d. of b 2 not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 0 Because we have replaced the standard deviation in its denominator with the standard error, the test statistic has a t distribution instead of a normal distribution. 3 5% significance test: reject H 0 : 2 = 2 if z > 1.96 or z < –1.96

4 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.: s.d. of b 2 not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 00 Accordingly, we refer to the test statistic as a t statistic. In other respects the test procedure is much the same. 4 5% significance test: reject H 0 : 2 = 2 if z > 1.96 or z < –1.96 5% significance test: reject H 0 : 2 = 2 if t > t crit or t < –t crit

5 5 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT We look up the critical value of t and if the t statistic is greater than it, positive or negative, we reject the null hypothesis. If it is not, we do not. s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : 2 = 2 if z > 1.96 or z < –1.96 s.d. of b 2 not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : 2 = 2 if t > t crit or t < –t crit 00

6 6 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Here is a graph of a normal distribution with zero mean and unit variance normal

7 7 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT A graph of a t distribution with 10 degrees of freedom (this term will be defined in a moment) has been added. normal t, 10 d.f.

8 8 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT When the number of degrees of freedom is large, the t distribution looks very much like a normal distribution (and as the number increases, it converges on one). normal t, 10 d.f.

9 9 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Even when the number of degrees of freedom is small, as in this case, the distributions are very similar. normal t, 10 d.f.

10 10 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Here is another t distribution, this time with only 5 degrees of freedom. It is still very similar to a normal distribution. normal t, 5 d.f. t, 10 d.f.

11 11 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT So why do we make such a fuss about referring to the t distribution rather than the normal distribution? Would it really matter if we always used 1.96 for the 5% test and 2.58 for the 1% test? normal t, 10 d.f. t, 5 d.f.

12 12 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The answer is that it does make a difference. Although the distributions are generally quite similar, the t distribution has longer tails than the normal distribution, the difference being the greater, the smaller the number of degrees of freedom. normal t, 10 d.f. t, 5 d.f.

13 13 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT As a consequence, the probability of obtaining a high test statistic on a pure chance basis is greater with a t distribution than with a normal distribution. normal t, 10 d.f. t, 5 d.f.

14 14 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT This means that the rejection regions have to start more standard deviations away from zero for a t distribution than for a normal distribution. normal t, 10 d.f. t, 5 d.f.

15 15 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The 2.5% tail of a normal distribution starts 1.96 standard deviations from its mean. normal t, 10 d.f. t, 5 d.f.

16 16 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The 2.5% tail of a t distribution with 10 degrees of freedom starts 2.33 standard deviations from its mean. normal t, 10 d.f. t, 5 d.f.

17 17 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT That for a t distribution with 5 degrees of freedom starts 2.57 standard deviations from its mean. normal t, 10 d.f. t, 5 d.f.

18 18 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT For this reason we need to refer to a table of critical values of t when performing significance tests on the coefficients of a regression equation. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% ………………… …………………

19 19 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT At the top of the table are listed possible significance levels for a test. For the time being we will be performing two-sided tests, so ignore the line for one-sided tests. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% ………………… …………………

20 20 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Hence if we are performing a (two-sided) 5% significance test, we should use the column thus indicated in the table. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% ………………… …………………

21 21 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT The left hand vertical column lists degrees of freedom. The number of degrees of freedom in a regression is defined to be the number of observations minus the number of parameters estimated. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% ………………… ………………… Number of degrees of freedom in a regression = number of observations – number of parameters estimated.

22 22 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT In a simple regression, we estimate just two parameters, the constant and the slope coefficient, so the number of degrees of freedom is n - 2 if there are n observations. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% ………………… …………………

23 23 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT If we were performing a regression with 20 observations, as in the price inflation/wage inflation example, the number of degrees of freedom would be 18 and the critical value of t for a 5% test would be t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% ………………… …………………

24 24 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT Note that as the number of degrees of freedom becomes large, the critical value converges on 1.96, the critical value for the normal distribution. This is because the t distribution converges on the normal distribution. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% ………………… …………………

25 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : 2 = 2 if z > 1.96 or z < –1.96 s.d. of b 2 not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : 2 = 2 if t > t crit or t < –t crit 00 Hence, referring back to the summary of the test procedure, 25

26 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : 2 = 2 if z > 1.96 or z < –1.96 s.d. of b 2 not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 5% significance test: reject H 0 : 2 = 2 if t > or t < – we should reject the null hypothesis if the absolute value of t is greater than

27 27 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT If instead we wished to perform a 1% significance test, we would use the column indicated above. Note that as the number of degrees of freedom becomes large, the critical value converges to 2.58, the critical value for the normal distribution. t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% ………………… …………………

28 28 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT For a simple regression with 20 observations, the critical value of t at the 1% level is t Distribution: Critical values of t Degrees of Two-sided test 10% 5% 2% 1% 0.2% 0.1% freedom One-sided test 5% 2.5% 1% 0.5% 0.1% 0.05% ………………… …………………

29 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT s.d. of b 2 known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 : 2 = 2 if z > 1.96 or z < –1.96 s.d. of b 2 not known discrepancy between hypothetical value and sample estimate, in terms of s.e.: 1% significance test: reject H 0 : 2 = 2 if t > or t < – So we should use this figure in the test procedure for a 1% test. 29

30 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT We will next consider an example of a t test. Suppose that you have data on p, the average rate of price inflation for the last 5 years, and w, the average rate of wage inflation, for a sample of 20 countries. It is reasonable to suppose that p is influenced by w. 30 Example:

31 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 31 Example: You might take as your null hypothesis that the rate of price inflation increases uniformly with wage inflation, in which case the true slope coefficient would be 1.

32 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 32 Example: Suppose that the regression result is as shown (standard errors in parentheses). Our actual estimate of the slope coefficient is only We will check whether we should reject the null hypothesis.

33 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 33 Example: We compute the t statistic by subtracting the hypothetical true value from the sample estimate and dividing by the standard error. It comes to –1.80.

34 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 34 Example: There are 20 observations in the sample. We have estimated 2 parameters, so there are 18 degrees of freedom.

35 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 35 Example: The critical value of t with 18 degrees of freedom is at the 5% level. The absolute value of the t statistic is less than this, so we do not reject the null hypothesis.

36 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 36 In practice it is unusual to have a feeling for the actual value of the coefficients. Very often the objective of the analysis is to demonstrate that Y is influenced by X, without having any specific prior notion of the actual coefficients of the relationship.

37 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 37 In this case it is usual to define 2 = 0 as the null hypothesis. In words, the null hypothesis is that X does not influence Y. We then try to demonstrate that the null hypothesis is false.

38 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 38 For the null hypothesis 2 = 0, the t statistic reduces to the estimate of the coefficient divided by its standard error.

39 t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 39 This ratio is commonly called the t statistic for the coefficient and it is automatically printed out as part of the regression results. To perform the test for a given significance level, we compare the t statistic directly with the critical value of t for that significance level.

40 . reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 40 Here is the output from the earnings function fitted in a previous slideshow, with the t statistics highlighted.

41 . reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 41 You can see that the t statistic for the coefficient of S is enormous. We would reject the null hypothesis that schooling does not affect earnings at the 0.1% significance level without even looking at the table of critical values of t.

42 . reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 42 The t statistic for the intercept is also enormous. However, since the intercept does not hve any meaning, it does not make sense to perform a t test on it.

43 . reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 43 The next column in the output gives what are known as the p values for each coefficient. This is the probability of obtaining the corresponding t statistic as a matter of chance, if the null hypothesis H 0 : = 0 is true.

44 . reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 44 If you reject the null hypothesis H 0 : = 0, this is the probability that you are making a mistake and making a Type I error. It therefore gives the significance level at which the null hypothesis would just be rejected.

45 . reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 45 If p = 0.05, the null hypothesis could just be rejected at the 5% level. If it were 0.01, it could just be rejected at the 1% level. If it were 0.001, it could just be rejected at the 0.1% level. This is assuming that you are using two-sided tests.

46 . reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 46 In the present case p = 0 to three decimal places for the coefficient of S. This means that we can reject the null hypothesis H 0 : 2 = 0 at the 0.1% level, without having to refer to the table of critical values of t. (Testing the intercept does not make sense in this regression.)

47 . reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT 47 It is a more informative approach to reporting the results of test and widely used in the medical literature. However in economics standard practice is to report results referring to 5% and 1% significance levels, and sometimes to the 0.1% level.

48 Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 2.6 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics


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